Multiply the following complex numbers, marked as blue dots on the graph: $[2(\cos(\frac{1}{4}\pi) + i \sin(\frac{1}{4}\pi))] \cdot [\cos(\frac{2}{3}\pi) + i \sin(\frac{2}{3}\pi)]$ (Your current answer will be plotted in orange.)
Answer: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $2(\cos(\frac{1}{4}\pi) + i \sin(\frac{1}{4}\pi))$ ) has angle $\frac{1}{4}\pi$ and radius $2$ The second number ( $\cos(\frac{2}{3}\pi) + i \sin(\frac{2}{3}\pi)$ ) has angle $\frac{2}{3}\pi$ and radius $1$ The radius of the result will be $2 \cdot 1$ , which is $2$ The angle of the result is $\frac{1}{4}\pi + \frac{2}{3}\pi = \frac{11}{12}\pi$ The radius of the result is $2$ and the angle of the result is $\frac{11}{12}\pi$.